Series: Penn State Logic Seminar Date: Thursday, November 29, 2001 Time: 9:45 - 11:00 AM Place: 169 Willard Building Speaker: Simon Thomas, Mathematics, Rutgers University Title: The Classification Problem for Torsion-Free Abelian Groups of Finite Rank Abstract: In 1937, Baer introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for the torsion-free abelian groups of rank 1. In this talk, I will discuss some recent contributions to the project of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n > 1. The talk will include a discussion of where this classification problem is situated within the descriptive set-theoretic context of countable Borel equivalence relations.